Algebraic Geometry and Singularities by Vincent Cossart (auth.), Antonio Campillo López, Luis

By Vincent Cossart (auth.), Antonio Campillo López, Luis Narváez Macarro (eds.)

The concentration of this quantity lies on singularity conception in algebraic geometry. It contains papers documenting fresh and unique advancements and strategies in topics similar to solution of singularities, D-module conception, singularities of maps and geometry of curves. The papers originate from the 3rd foreign convention on Algebraic Geometry held in l. a. Rábida, Spain, in December 1991. on the grounds that then, the articles have passed through a meticulous strategy of refereeing and development, they usually were equipped right into a finished account of the state-of-the-art during this field.

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X to y). A point x is said to be dominant if (y, x) E e for any y E x. Acyclic graphs have some nice properties. First, for an acyclic graph there exists at least one point x (resp. y) such that x = {x} (resp. y = {y}). Such a point will be said to be a minimal (resp. maximal) in the graph. foreover, the points in an acyclic graph (X, G) can be distributed by levels No, N I , ... e e e e. I\',)}. , the greatest index such that Nk i- 0. Second, we have the following characterization of acyclic graphs (see [3] for details): A graph (X, G).

0 7rl, and such that the strict transform XN of X at ZN is smooth and transversal (normal crossing) to the exceptional divisor EN. The embedded resolution of singularities was extended to analytical spaces in 1974 in [IJ. 1) for each variety X, This leads us to consider an algorithmic point of view (see [5]). Looking for such an algorithmic viewpoint, our initial motivation, in this paper, is to try to understand the behaviour, after blowing up 7ri+l, of the set (or a subset) of subvarieties Ei created at Zi by the sequence of transformations, More precisely, we are interested in the following question: if 7r: Z' --+ Z is a blowing up of a smooth variety at a regular center C, and if C is a smooth geometrical configuration of subvarieties of Z, try to describe the transform C' of C in Z', i,e" the set of strict transforms of subvarieties in C not included in C and the fibers of those contained in C, as well as the irreducible components of the intersections of such objects.

Si x = I: aiti, Y = I: biti , Z = I: Citi est la parametrisation d'une courbe tracee sur (S,O) et e est sa multiplicite, les conditions suivantes caracterisent son appartenance it 1t2 '1' 1 ::::: i ::::: 4 : 1t221 1t2I uf+br = 0 al =f. 0 (' = 2~~t~~~ = () I 1t23I I (. ~ 2, (~3+~~:('~ = 0 1{24 1 ( ~ :3, a:;+bi = () a4 =f. 0 La transformee stricte sur SI d'une courbe generale tracee sur (5,0) appartenant it 1{2'l, 1 ::::: i ::::: 4, est une courbe lisse et transverse a d 1 ou d~ en Q =f. d l nd~ si i = 1, ayant un contact d'ordre 2 avec d l ou d~ en Q =f.

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